Every Student Learns … Every Day!

There was a period in my teaching when the core principle was “deep assessment”.  This “Deep Assessment” idea was that every key outcome within a test would be assessed three times BEFORE the test for each student, in class … at the intro level when starting the topic, at an intermediate level after the first usage, and at a higher level as part of the review for the test.  I would tell my colleagues that I assessed the important ideas 3 times, and they seemed to think this was good … and so did I, until I thought about my observations.

Sure, it helps students to have multiple opportunities (assessments) on key ideas and get instructor feedback.  I would spend considerable time grading these assessments, and writing feedback.  This very logical structure did, in fact, work for a portion of my students.  As I thought about this, however, most of the students who benefited were doing fairly well before my class.  You know, they were mostly reviewing stuff they once knew well.

However, the ‘deep assessment’ strategy missed some of the students in the middle (of need), and missed almost all of the students with the greatest need.  Do our classes exist to serve just some students, or all?  Hopefully, you think about that question on a regular basis.  There are direct connections between that question and the posts made recently about equity Policy based on Correlation: Institutionalizing Inequity.

My current guiding principle is “everybody learns every day”.  I seek to provide some benefit to all students.

Many readers are going to be thinking … “What’s so different about that? Don’t we provide the opportunity to benefit every student in each class?”

Nope, we don’t.  Think about this … “learning” depends upon readiness and engagement, combined with communication.  We fail to address the readiness almost all of the time.  By this I am not referring to course prerequisites or placement tests; those are gross measures of overall abilities, and have very little to do with learning.

I’m referring to a thorough analysis of specific knowledge and understanding needed to learn a certain topic.  Let’s look at “basic function ideas” as you might cover in an intermediate algebra or college algebra course.  Learning basic function ideas (notation, interpretation, points) at an introductory level.  The readiness includes:

• input versus output
• simplifying expressions
• substituting values
• horizontal and vertical number lines
• ordered pair notation and meaning
• point plotting as opposed to slope

The image above shows ‘puzzle pieces’ between the person and the learning.  Vygotsky used a phrase “zone of proximal development”, which is related to what I am talking about.  [Vygotsky was primarily a developmental psychologist, so his results are indirectly related to current learning sciences.]

The ‘ready to learn’ criteria is always there.  If we ignore it, we only serve part of the students.  On the other hand, if we tell students that they need to ‘review’ something before the new stuff, we expect the weakest students to do the more complicated process without our direct support and advice.

I’m teaching developmental math, not ‘college level’, so my dive into this is really intense.  Every class day, we start with a team activity which both checks on the readiness and begins the process of learning today’s stuff.  We might spend 20 minutes doing the activity, followed by 10 minutes of reviewing it as a class; my goal is to get everybody ready, and have everybody learn every day.  Small teams (3 to 5) does a pretty good job of keeping everybody involved, and making sure that everybody is learning.

In a college level course, we could still use a team activity on readiness.  Depending on the topic, we might only need 10 minutes doing it, and 5 minutes reviewing it.  In other cases, the ‘readiness to learn’ activity might occupy the majority of the class time.

I can’t tell you that my ‘plan’ is perfect; that’s a unreasonably high standard (even for me 🙂 ).  However, I can tell you that this “everybody learns everyday” approach does wonders for attendance and participation.  My students with the greatest need still have gaps, but they are smaller.  The ‘middle’ students tend to look more like the high-quality (reviewing) students.

We know that ‘attendance’ is highly correlated with success in mathematics.  Students with greater learning needs get easily discouraged when our classes do not provide them with much learning — either due to lack of readiness (at the detailed level) OR due to our class structure not engaging every learner.  “Everybody learns everyday” minimizes this systemic risk, without harming the higher achieving students.

Policy based on Correlation: Institutionalizing Inequity

The higher-ed world (in the United States) is all aglow with “pathways”, both in the sense of program pathways with limited options and in the sense of mathematics courses.  Both trends are heavily based on the use (and mis-use) of statistics, and both will tend to reinforce social inequities.  Let’s explore “what’s wrong” with the policies hard-coded in financial aid guidelines about ‘courses not on the Program’ and the local & regional policies of “college math course in the first year”.

[Pell Institute data]

First, we must recognize the increasing levels of inequity in the United States. Those with wealth or higher incomes are likely to complete college degrees, so the upper class children tend to have wealth & income equal to their parents (or higher).  Those with less wealth and less income are less likely to complete, and are much more likely to do so with significant student loan burdens.

As a result of this and similar statistics, federal financial aid is demanding that students only take courses on their program … based on the logical presumption that this will tend to result in more degree completions (compared to students taking courses not counting for a degree).  All students are subject to this policy, which is an application of ‘equality’.  Take a look at this image:

Whenever we apply equality to a situation with inequity, we tend to increase that inequity.  Consider three prototypical students that we have in our institutions:

• Eberle comes from a high-performing K-12 system, and a family with significant resources.  He’s known what his college degree was going to be for ten years.
• Tykese comes from a low-performing K-12 system in the city, and a family with economic insecurity.  He did not even consider attending a college until earlier this year.
• Kylen comes from a typical rural K-12 system with limited resources, and a family with economic challenges.  He has been planning on attending college for three years, but is unsure whether to do “STEM” or business.

In the image above, Eberle is on the tall pile of boxes; he’s got a built-in advantage, and he will succeed (often in spite of his professors).  The requirement to only take courses on the program will seem like a no-brainer to Eberle.  Of course … what else would I do?

Tykese is the guy who can’t even see over the fence (‘reality’).  His attending college is fragile, and he lacks a solid goal when he starts.  He needs the equity image, where he is given the support to explore some options prior to being locked in to a program.  What he gets is a demand to select a program and only take courses listed on it; in the equity image, he still does not have a fair chance.  Tykese is likely to drop out, with student loan debt.

Kylen could be the guy in the middle of the images; he could also be the guy on the right.  The rural settings can produce over-achieving students (left guy) or folks resembling the inequity guy (right).  However, the demand to claim a program and stick to it results in additional academic pressure.

In a recent post, I wrote about ‘white privilege’ Gift as Responsibility: White Privilege.  Some readers might have thought that I was advocating equality in a math classroom.  I, personally, do not seek to treat students all the same … I strive for equity, understanding that this goal is not totally possible.  An approach which says “I give all students the same” reflects an acceptance of the status quo; if there was not social inequity  in our country and our region, this ‘same’ approach would make sense.  In an era of increasing inequity, equality is mismatched to our educational mission.

[Cartoon related to school funding in Pennsylvania … though the pattern is not limited to one state or region.]

The ‘courses must be in the program’ rule is based on a goal to get more students to complete, and is a policy  of ‘equality’.  In reality, therefore, the financial aid rules are tending to reinforced the existing inequity.  For community colleges in particular, this approach hurts the very students most in need of our institutions.

So, on to math pathways.  Coupled with the financial aid rules, there is a strong push to get students to complete their credit math course in the first year.  I’ve written on some of the issues Why Do Students Have to Take Math in College? Today, I am thinking more of the interaction between the “first year” policy combined with the ‘program courses only’ rule.

So, let’s go back to Tykese.  He comes to college orientation, and meets with an advisor to figure out what he should do.  The advisor notices the overall lack of clear goal, and suggests that Tykese consider programs with an ‘easier’ math requirement.  There is a ‘tracking’ element happening here, where students with weak academic backgrounds are directed towards lower level programs; this would be fine if the credential for these programs led to long-term employment at a decent salary.

In most cases, however, the programs with lower level math requirements tend to be associated with weak employment prospects.  The exceptions — primarily in health careers — have significant demands for science courses and high academic performance.  Tykese is also at a disadvantage in science.

To make this a reasonable 2019 story, Tykese would be placed directly into a quantitative reasoning course with a required ‘corequisite’ course.  Most of the other students in this ‘blue bird’ corequisite class are students of inequity like Tykese; segregation is alive and well.  Few students like Tykese get to STEM oriented math courses, because their algebra background is poor and we don’t provide equity in their mathematics.

The motivation for the ‘first year’ rule about math is pretty much limited to correlational data: students who passed a math course in the first year are more likely to complete their degree.   In other words, students like Eberle complete degrees and complete college math in the first year; requiring students to take college math in the first year will not magically create students who will complete their program.  Students in the lower SES levels will still face the challenges making it difficult to complete; all we’ve achieved is to push the weaker students to take ‘easier’ math courses.

The ‘math in the first year’ functions as a rationale to track students based on where they happened to go to school.

I hope that you see equity as your calling, as I do.  All of our students deserve access to economic security as well as a broad education.  Rules like “courses must be on the program” and policies like “credit math course in the first year” reinforce existing inequity.

[By the way, all three ‘students’ listed are based on recent students in my classes; all three were ‘black’, though I’ve adjusted the biographical sketch for each of them.]

Gift as Responsibility: White Privilege

This post is about white privilege, on a blog devoted to renewal of college mathematics.  What are the connections?  Can we save college mathematics without understanding the role of white privilege in higher education?  Do you accept the existence of ‘privilege’?

I think about white privilege, and the role it played in my own journey, on a regular basis.  The motivation to make this post comes from conversations we are having in my department about addressing the equity gap — the differential outcomes for specific ‘ethnic’ groups, specifically black students.  This past week, a colleague suggested that she could begin to build better relationships by requiring every student in the class to speak with her outside of class.  Like me, this instructor is white.

Students with a privileged condition will have a reasonable choice about doing something like talking with the instructor outside of class.  Students with low privilege, no privilege, or negative privilege will face some challenges in complying with this directive from a white instructor.  You might be thinking that all students would try to avoid talking with a math instructor privately, and there is some truth in the statement.  However, those of us who have benefited from white privilege often fail to understand the range of forces involved in low- and high-privilege situations.

If you are white in America, you may in fact wonder if privilege exists … perhaps people with status and power have worked harder and were smarter.  Here is a one question ‘test’ to see if you have privilege relative to ethnicity and race:

Do you have the choice to control how much your identity is based on ethnicity and/or race?

Some people I know are white, and some of them will say that they don’t see themselves as white; that is a clear statement of white privilege.  I’ve yet to hear a black person say that they don’t see themselves as black; sure, some “black folks” identify very strongly as black, and others less so.  The point is … in America, being black means that you don’t control your basic identity.

Okay, so accept (perhaps for the sake of argument) that white privilege exists.  Does it have anything to do with higher education, in mathematics specifically?

To answer that question, just think about the most common observations about ‘struggling’ students:

• Passive
• Not prepared
• Lack of academic facility

How do students arrive in our classes ready to be active, to be prepared, and to have academic facility?  They had a combination of ‘family’ and K-12 education that provided a basis for functioning in a mathematics classroom.  A student with white privilege is more likely to have had parents with the time and inclination to actively support learning … and support risk taking.  A student with white privilege is more likely to have attended a K-12 system with opportunities for higher levels of learning, as well as better preparation for college.

White privilege is a gift that I received at birth, paid for by predecessors choices; much of my white privilege is the direct and indirect result of inhuman treatment … racism … and wealth accumulation.  I am not responsible for the gift, but I am responsible for being able to use that gift.  I can go through life pushing to higher levels of power and wealth, or I can use the gift as little as possible.  I can’t avoid using the gift — when I stand in front of a class, speak to my administrators, or even post on this blog, the gift of white privilege is there giving me an advantage.  Every time I use this gift of white privilege, somebody else is prevented from building a privileged condition for themselves.

I can’t speak for other  ‘white folk’, just like a black colleague not being able to speak for ‘black folk’.

However, I intend to continue using this gift of privilege as little as possible.  I seek to lift up those who did not get this same gift, even when it means some loss to me.  I don’t claim to be very good at that; in fact, I would judge that I have missed far too many opportunities to increase the privileged condition for those around me … and those in my classrooms specifically.

In my classrooms, I do not need to do much to get higher status.  Therefore, I can spend more of my energy — especially in the first week of class — on creating an inclusive classroom.  An inclusive classroom is “privilege proof”, meaning that the benefits (learning) have no connection to the level of privilege for any student.  Throughout the semester, I need to continue to include and build up all students; students who lack privilege previously should get more of my support.

If that last part bothers you (about differential support), just think about this:  Students with the most ‘white privilege’ do not need us for much of anything.  They will be successful learning mathematics without us, or even in spite of us.  An unskilled or uncaring instructor will harm low privilege students, but will have almost no impact on students like me — because of the white privilege.

White privilege is especially important in school settings.  In higher education, we once had a system which only served those with privilege … almost universally white males in this country.  In my view, if you are not thinking about white privilege, you can only be an effective educator if all your students are ‘just like you’ (in terms of race/ethnicity).

White privilege provides a systemic advantage to a specific group of people.  If you are a member of this group, I think the gift of white privilege imposes upon you a responsibility to consciously minimize the use of the gift of privilege.  The use of privilege will reinforce the lack of privilege for another group — every time privilege is used.

Pre-Calculus, Rigor and Identities

Our department is working on some curricular projects involving both developmental algebra and pre-calculus.  This work has involved some discussion of what “rigor” means, and has increased the level of conversation about algebra in general.  I’ve posted before about pre-calculus College Algebra is Not Pre-Calculus, and Neither is Pre-calc and College Algebra is Still Not Pre-Calculus 🙁 for example, so this post will not be a repeat of that content. This post will deal with algebraic identities.

So, our faculty offices are in an “open style”; you might call them cubicles.  The walls include white board space, and we have spaces for collaboration and other work.  Next to my office is a separate table, which one of my colleagues uses routinely for grading exams and projects.  Recently, he was grading pre-calculus exams … since he is heavily invested in calculus, he was especially concerned about errors students were making in their algebra.  Whether out of frustration or creative analysis, he wrote on the white board next to the table.  Here is the ‘blog post’ he made:

This picture is not very readable, but you can probably see the title “Teach algebraic identities”, followed by “Example:  Which of the following are true for all a, b ∈ ℜ.  In our conversation, my colleague suggested that some (perhaps all) of these identities should be part of a developmental algebra course.  The mathematician part of my brain said “of course!”, and we had a great conversation about the reasons some of the non-identities on the list are so resistant to correction and learning.

Here are images of each column in the post:

When we use the word “identities” in early college mathematics, most of us expect the qualifier to be “trig” … not “algebraic”.  I think we focus way too much on trig identities in preparation for calculus and not enough on algebraic identities.  The two are, of course, connected to the extent that algebraic identities are sometimes used to prove or derive a trig identity.  We can not develop rigor in our students, including sound mathematical reasoning, without some attention to algebraic identities.

I think this work with algebraic identities begins in developmental algebra.  Within my own classes, I will frequently tell my students:

It is better for you to not do something you could … than to make the mistake of doing something ‘bad’ (erroneous reasoning).

Although I’ve not used the word identities when I say this, I could easily phrase it that way: “Avoid violating algebraic identities.”  Obviously, few students know specifically what I mean at the time I make these statements (though I try to push the conversation in class to uncover ‘bad’, and use that to help them understand what is meant).  The issue I need to deal with is “How formal should I make our work with algebraic identities?” in my class.

I hope you take a few minutes to look at the 10 ‘identities’ in those pictures.  You’ve seen them before — both the ones that are true, and the ones that students tend to use in spite of being false.  They are all forms of distributing one operation over another.  When my colleague and I were discussing this, my analysis was that these identities were related to the precedence of operations, and that students get in to trouble because they depend on “PEMDAS” instead of understanding precedence (see PEMDAS and other lies 🙂 and More on the Evils of PEMDAS!   ).  In cognitive science research on mathematics, the these non-identities are labeled “universal linearity” where the basic distributive identity (linear) is generalized to the universe of situations with two operations of different precedence.

How do we balance the theory (such as identities) with the procedural (computation)?  We certainly don’t want any mathematics course to be exclusively one or the other.  I’m envisioning a two-dimensional space, where the horizontal axis if procedural and the vertical axis is theory.  All math courses should be in quadrant one (both values positive); my worry is that some course are in quadrant IV (negative on theory).  I don’t know how we would quantify the concepts on these axes, so imagine that the ordered pairs are in the form (p, t) where p has domain [-10, 10] and t has range [-10, 10].  Recognizing that we have limited resources in classes, we might even impose a constraint on the sum … say 15.

With that in mind, here are sample ordered pairs for this curricular space:

• Developmental algebra = (8, 3)           Some rigor, but more emphasis on procedure and computation
• Pre-calculus = (6, 8)                         More rigor, with almost equal balance … slightly higher on theory
• Calculus I to III = (5, 10)                   Stronger on rigor and theory, with less emphasis on computation

Here is my assessment of traditional mathematics courses:

• Developmental algebra … (9, -2)         Exclusively procedure and computation, negative impact on theory and rigor
• Pre-calculus … (10, 1)                           Procedure and computation, ‘theory’ seen as a way to weed out ‘unprepared’ students
• Calculus I to III … (10, 3)                      A bit more rigor, often implemented to weed out students who are not yet prepared to be engineers

Don’t misunderstand me … I don’t think we need to “halve” our procedural work in calculus; perhaps this scale is logarithmic … perhaps some other non-linear scale.  I don’t intend to suggest that the measures are “ratio” (in the terminology of statistics; see  https://www.questionpro.com/blog/ratio-scale/ ).  Consider the measurement scales to be ordinal in nature.

I think it is our use of the ‘theory dimension’ that hurts students; we tend to either not help students with theory or to use theory as a way to prevent students from passing mathematics.  The tragedy is that a higher emphasis on theory could enable a larger and more diverse set of students to succeed in mathematics, as ‘rigor’ allows other cognitive strengths to help a student succeed.  The procedural emphasis favors novice students who can remember sequences of steps and appropriate clues for when to use them … a theory emphasis favors students who can think conceptually and have verbal skills; this shift towards higher levels of rigor also serves our own interests in retaining more students in the STEM pipeline.